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Math Help - Agoh's Conjecture

  1. #1
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    Agoh's Conjecture

    Hi everyone

    I encountered Agoh's Conjecture:

    <br /> <br />
nB_{n-1}\equiv-1\quad mod \quad n<br /> <br />

    When n is prime.

    Bn is the n th Bernoulli numer.

    How can we extract modulo from fractions?
    I red somewhere about minimal residue but i did not understand exacty what it is.
    Can you give me some examples (modulo of fractions)?

    Thank you for your time
    Last edited by gdmath; August 4th 2009 at 02:31 AM. Reason: Incomplete
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by gdmath View Post
    Hi everyone

    I encountered Agoh's Conjecture:
    <br />
nB_n\equiv-1 \mod n<br />
    Is this a question?

    (Presumably B_n denotes the n -th Bernoulli number, you should say so)

    Note the conjecture is for n prime

    CB
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  3. #3
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    Sorry i accidently posted the thread without complete it.
    The whole statement is:

    "
    Hi everyone

    I encountered Agoh's Conjecture:



    When n is prime.

    Bn is the n th Bernoulli numer.

    How can we extract modulo from fractions?
    I red somewhere about minimal residue but i did not understand exacty what it is.
    Can you give me some examples (modulo of fractions)?

    Thank you for your time
    Follow Math Help Forum on Facebook and Google+

  4. #4
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    Quote Originally Posted by gdmath View Post
    Sorry i accidently posted the thread without complete it.
    The whole statement is:

    "
    Hi everyone

    I encountered Agoh's Conjecture:



    When n is prime.

    Bn is the n th Bernoulli numer.

    How can we extract modulo from fractions?
    I red somewhere about minimal residue but i did not understand exacty what it is.
    Can you give me some examples (modulo of fractions)?

    Thank you for your time

    suppose  n = 19

    The 18th Bernoulli Number


     B_{18} =  \dfrac{43867}{798}



    from the equation:  nB_{n-1}

     19 \cdot \dfrac{43867}{798} REDUCES TO  \dfrac{43867}{42}

    The modular inverse of 42 with modulus 19 is 5.

    43867 x 5 = 219335

     219335 \, \equiv \, 18 \, mod(19) \, \equiv \, -1 \, mod(19)

    The fractions are handled by using the modular inverse.

    FYI:
    The one millionth Bernoulli number has more than 4.7 million digits in the numerator.
    &
    The two millionth Bernoulli number has more than 10 million digits in the numerator.

    for additional information see: Kellner, B. C. The Equivalence of Giuga’s and Agoh’s Conjectures. 15 Sep 2004.
    which can be found here: [math/0409259] The Equivalence of Giuga's and Agoh's Conjectures.
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